International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

International Tables for Crystallography (2006). Vol. C. ch. 4.2, pp. 251-252

Section 4.2.6.3.3.1. Measurements in the high-energy limit [(\omega/\omega_\kappa\rightarrow0)]

D. C. Creaghb

4.2.6.3.3.1. Measurements in the high-energy limit [(\omega/\omega_\kappa\rightarrow0)]

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In this case, there is some possibility of testing the validity of the relativistic dipole and relativistic multipole theories since, in the high-energy limit, the value of [f'(\omega,0)] must approach a value related to the total self energy of the atom [(E_{\rm tot}/mc^2)]. That there is an atomic number dependent systematic error in the relativistic dipole approach has been demonstrated by Creagh (1984[link]). The question of whether the relativistic multipole approach yields a result in better accord with the experimental data is answered in Table 4.2.6.4[link], where a comparison of values of [f'(\omega,0)] is made for three theoretical data sets (this work; Cromer & Liberman, 1981[link]; Wagenfeld, 1975[link]) with a number of experimental results. These include the `direct' measurements using X-ray interferometers (Cusatis & Hart, 1975[link]; Creagh, 1984[link]), the Kramers–Kronig integration of X-ray attenuation data (Gerward et al., 1979[link]), and the angle-of-the-prism data of Deutsch & Hart (1984b[link]). Also included in the table are `indirect' measurements: those of Price et al. (1978[link]), based on Pendellösung measurements, and those of Grimvall & Persson (1969[link]). These latter data estimate [f'(\omega,{\bf g}_{hkl})] and not [f'(\omega,0)]. Table 4.2.6.4[link] details values of the real part of the dispersion correction for LiF, Si, Al and Ge for the characteristic wavelengths Ag [K\alpha_1], Mo [K\alpha_1] and Cu [K\alpha_1]. Of the atomic species listed, the first three are approaching the high-energy limit at Ag [K\alpha_1], whilst for germanium the K-shell absorption edge lies between Mo [K\alpha_1] and Ag [K\alpha_1].

Table 4.2.6.4| top | pdf |
Comparison of measurements of the real part of the dispersion correction for LiF, Si, Al and Ge for characteristic wavelengths Ag Kα1, Mo Kα1 and Cu Kα1 with theoretical predictions; the experimental accuracy claimed for the experiments is shown thus: (10) = 10% error

SampleReference[f'(\omega,0)]
Cu [K\alpha_1]Mo [K\alpha_1]Ag [K\alpha_1]
LiFTheory   
 This work0.0750.0170.010
 Cromer & Liberman (1981[link])0.0680.0140.006
 Wagenfeld (1975[link])0.0800.0230.015
Experiment   
 Creagh (1984[link])0.085 (5)0.020 (10)0.014 (10)
 Deutsch & Hart (1984b)0.0217 (1)0.0133 (1)
SiTheory   
 This work0.2540.8170.052
 Cromer & Liberman (1981[link])0.2420.0710.042
 Wagenfeld (1975[link])0.2820.1010.071
Experiment   
 Cusatis & Hart (1975[link])0.0863 (2)0.0568 (2)
 Price et al. (1978[link])0.085 (7)0.047 (7)
 Gerward et al. (1979[link])0.244 (7)0.099 (7)0.070 (7)
 Creagh (1984[link])0.236 (5)0.091 (5)0.060 (5)
 Deutsch & Hart (1984b[link])0.0847 (1)0.0537 (1)
AlTheory   
 This work0.2130.06450.041
 Cromer & Liberman (1981[link])0.2030.04860.020
 Wagenfeld (1975[link])0.2350.0760.553
Experiment   
 Creagh (1985[link])0.065 (20)0.044 (20)
 Takama et al. (1982[link])0.20 (5)0.07 (5)0.035 (10)
GeTheory   
 This work−1.0890.1550.302
 Cromer & Liberman (1981[link])−1.1670.0620.197
 Wagenfeld (1975[link])−1.80−0.080.14
Experiment   
 Gerward et al. (1979[link])−1.040.300.43
 Grimvall & Persson (1969[link])−1.790.080.27

The high-energy-limit case is considered first: both the relativistic dipole and relativistic multipole theories underestimate [f'(\omega,0)] for LiF whereas the non-relativistic theory overestimates [f'(\omega,0)] when compared with the experimental data. For silicon, however, the relativistic multipole yields values in good agreement with experiment. Further, the values derived from the work of Takama et al. (1982[link]), who used a Pendellösung technique to measure the atomic form factor of aluminium are in reasonable agreement with the relativistic multipole approach. Also, some relatively imprecise measurements by Creagh (1985[link]) are in better accordance with the relativistic multipole values than with the relativistic dipole values.

Further from the high-energy limit (smaller values of [\omega/\omega_\kappa)], the relativistic multipole approach appears to give better agreement with theory. It must be reported here that measurements by Katoh et al. (1985a[link]) for lithium fluoride at a wavelength of 0.77366 Å yielded a value of 0.018 in good agreement with the relativistic multipole value 0.017.

At still smaller values of [(\omega/\omega_\kappa)], the non-relativistic theory yields values considerably at variance with the experimental data, except for the case of LiF using Cu [K\alpha_1] radiation. The relativistic multipole approach seems, in general, to be a little better than the relativistic approach, although agreement between experiment and theory is not at all good for germanium. Neither of the experiments cited here, however, has claims to high accuracy.

In Table 4.2.6.5[link], a comparison is made of measurements of [f''(\omega,0)] derived from the results of the IUCr X-ray Attenuation Project (Creagh & Hubbell, 1987[link], 1990[link]) with a number of theoretical predictions. The measurements were made on carbon, silicon and copper specimens at the characteristic wavelengths Cu [K\alpha_1], Mo [K\alpha_1] and Ag [K\alpha_1]. The principal conclusion that can be drawn from perusal of Table 4.2.6.5[link] is that only minor, non-systematic differences exist between the predictions of the several relativistic approaches and the experimental results. In contrast, the non-relativistic theory fails for higher values of atomic number.

Table 4.2.6.5| top | pdf |
Comparison of measurements of f′(ω, 0) for C, Si and Cu for characteristic wavelengths Ag Kα1, Mo Kα1 and Cu Kα1 with theoretical predictions; the measurements are from the IUCr X-ray Attenuation Project Report (Creagh & Hubbell, 1987[link], 1990[link]), corrected for the effects of Compton, Laue–Bragg, and small-angle scattering

SampleReference[f'(\omega,0)]
Cu [K\alpha_1]Mo [K\alpha_1]Ag [K\alpha_1]
6CTheory   
 This work0.00910.00160.0009
 Cromer & Liberman (1981[link])0.00910.00160.0009
 Wagenfeld (1975[link])
 Scofield (1973[link])0.00930.00160.0009
 Storm & Israel (1970[link])0.00900.00160.0009
Experiment   
 IUCr Project0.00930.00160.0009
14SiTheory   
 This work0.3300.0700.043
 Cromer & Liberman (1981[link])0.3300.07040.0431
 Wagenfeld (1975[link])0.3300.0710.044
 Scofield (1973[link])0.3320.07020.0431
 Storm & Israel (1970[link])0.3310.06980.0429
Experiment   
 IUCr Project0.3320.06960.0429
29CuTheory   
 This work0.5881.2650.826
 Cromer & Liberman (1981[link])0.5891.2650.826
 Scofield (1973[link])0.5861.2560.826
Experiment   
 IUCr Project0.5881.2670.826

References

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